236 XIII INTEGRATION integrals Jn = f vn(t) dp(t) is bounded above. For this purpose, note that we may write vn(t) = (j/(t)to)(0)/wC/(0) for a suitably chosen /^-measurable mapping j of X into I, and majorize Jtf by using the Cauchy-Schwarz inequality and (a) above.) (c) Suppose that the hypotheses of (b) are satisfied and also that lim w(ri) = +00. 00 OO If the an are real and if the series £ #« w2W converges, show that the series £ anfn(t) n=l ir-l converges almost everywhere in A. (Start by using Problem 8(b) of Section 13.11. Then, in order to majorize \sn(t) — snk(t)\ for nk^n<nk+lt determine an increasing sequence (cn) of numbers >0 such that limc^+oo and £ tf w2(/z)c* < + oo (Section 5.3, Problem 6), so that the numbers bn = an w(ri)cn are of the form (g \fn) for some g e -^R(X, ft). Then use the fact that, by virtue of (b), the partial sums are bounded for almost all / e A, and apply Abel's partial summation *_i formula.) (d) Deduce from (c) that if |HnCs)| ^ c for all s e A and all «, and if the series oo converges, then the series Y< anfn(t) converges almost everywhere in A. (Use Problem n=l 6 of Section 5.3 again.) CO 12. (a) Let (cn) be a sequence of real numbers such that £ cj < +00. Show that the n=l oo series £ cnrn(t) (the notation is that of Problem 10) converges almost everywhere in I with respect to Lebesgue measure A. (Express the Rademacher functions as linear combinations of the functions of the Haar orthonrmal system (Section 8.7, Problem 7), and note that Problem 1 Id) is applicable to the latter orthonormal system (cf. Section 13.17, Problem 2).) (b) Given any sequence (an) of real numbers, a real e > 0 and a measurable set A c I, show that there exists an integer «0 such that whenever m>n^nQ. (Use the Cauchy-Schwarz inequality, the fact that the functions riMo(0 with '<7 form an orthonormal system (Problem 10(d)) and BesseFs in- equality applied to this system and to the function <pA , (c) Let (amn) be a double sequence of nonnegative numbers such that for each m the set of integers w ^ 1 for which amn ^ 0 is finite, and such that lim amn = 1 for each m-^oo n. Also let (cn) be a sequence of complex numbers such that, if we put the sequence (Sm(f)) is convergent in an integrable set A c I of positive measure. CO Show that X | c« 1 2 < + °° • (By virtue of Egoroff 's theorem, there exists an integer n=l n0 and a set B c A of positive measure, such that for all m^nQ and q > p ^ HQ , we I. For each /el, the set TT"!(/)